Question

Refer to the Interpreting Confidence Intervals applet. a. Suppose that you have a random sample of size \(n=50\) from a population with unknown mean \(\mu\) and known standard deviation \(\sigma=35 .\) Calculate the half width of a \(95 \%\) confidence interval for \(\mu\). What would the width of this interval be? b. Use the button to create a single confidence interval for \(\mu\). What is the width of this interval? Compare your results to the calculation you did in part a.

Short answer

Question: Calculate the half width and the full width of a 95% confidence interval for the population mean, given a sample size of 50 and a population standard deviation of 35. Compare your calculations to a single confidence interval obtained using the Interpreting Confidence Intervals applet. Answer: The half-width of the 95% confidence interval is approximately 9.702 and the full width of the 95% confidence interval is approximately 19.404. To compare these results to a single confidence interval created using the Interpreting Confidence Intervals applet, you would need to access the applet and create a single confidence interval for the population mean, and then compare the widths of that interval with the calculated values. The intervals' widths should be similar, but not exactly the same due to the randomness in drawing samples from different populations.

Step-by-step solution

Identify the given information

We are given the following information: - Sample size: \(n = 50\) - Population standard deviation: \(\sigma = 35\) We want to calculate the half width of a 95% confidence interval for the population mean and its full width.

Calculate the half-width of the 95% confidence interval

To calculate the half width of a 95% confidence interval for the population mean, we can use the following formula: Half-width \(= z \cdot \frac{\sigma}{\sqrt{n}}\) Where \(z\) is the z-score corresponding to the desired confidence level (95%), \(\sigma\) is the population standard deviation, and \(n\) is the sample size. For a 95% confidence interval, the z-score is approximately 1.96. Now, we can plug in the given values into the formula: Half-width \(= 1.96 \cdot \frac{35}{\sqrt{50}}\) Half-width \(= 1.96 \cdot 4.949\) (approximately) Half-width \(= 9.702\) (approximately) So, the half-width of the 95% confidence interval is approximately 9.702.

Calculate the full width of the 95% confidence interval

Now that we have the half width, we can calculate the full width of the confidence interval by simply multiplying the half width by 2: Width \(= 2 \cdot\) half-width Width \(= 2 \cdot 9.702\) Width \(= 19.404\) (approximately) So, the full width of the 95% confidence interval is approximately 19.404.

Compare the results with the Interpreting Confidence Intervals applet

We do not have access to the applet here, so we cannot create a single confidence interval using the applet at this time. However, you can refer to the Interpreting Confidence Intervals applet, create a single confidence interval for the population mean, and compare the width of that interval with the 19.404 (approximately) calculated in Step 3. Observe that the intervals' widths should be similar, but may not be exactly the same due to the randomness in drawing samples from different populations.

Key concepts

Confidence Interval Width

The width of a confidence interval tells us the range within which our true population parameter, such as the mean \(\mu\), is expected to lie with a particular level of confidence. In order to calculate this, we multiply the confidence interval's margin of error by two. The margin of error is the product of the z-score for the given confidence level and the standard error of the sample mean, the latter being the population standard deviation (\(\sigma\)) divided by the square root of the sample size (\(n\)).

For example, with a 95% confidence level, the z-score is approximately 1.96. If our population standard deviation is 35 and our sample size is 50, the width of our confidence interval around the sample mean is approximately 19.404, meaning we can be 95% confident that the true mean lies within this range from our sample mean.

The interpretation is crucial; it is not that the true mean will definitely be within this range, but rather that we have a high level of confidence based on our sample data, and if we were to take many samples, 95% of the calculated intervals would contain the true mean.

Sample Size

The sample size, denoted as \(n\), significantly affects the confidence interval's width. As the sample size increases, the margin of error decreases, resulting in a narrower confidence interval. This happens because a larger sample more closely approximates the whole population, reducing the standard error, which is directly related to the width of the confidence interval.

It is important to note that the relationship between sample size and interval width is not linear; rather, the standard error is inversely proportional to the square root of the sample size. To cut the width in half, you would need to quadruple the sample size. Understanding the relationship between sample size and interval width can help researchers determine the number of observations needed to achieve a particular precision in their estimates.

Population Standard Deviation

Population standard deviation (\(\sigma\)) is a measure of variability or spread within a data set. It plays a pivotal role in calculating the width of a confidence interval, as it's used to determine the standard error of the sample mean. The standard error is \(\sigma\) divided by the square root of the sample size (\(n\)), and this term is then multiplied by the z-score to give the margin of error. A higher population standard deviation indicates more variability in the data and, therefore, results in a wider confidence interval assuming the sample size and confidence level remain constant.

Knowing the population standard deviation is especially important in cases where it can be assumed to be known or is estimated with high accuracy from past data. This assumption is used in creating confidence intervals for the mean when the underlying distribution is normal or when sample sizes are large enough for the Central Limit Theorem to hold.

z-score

The z-score is a statistical measure that tells us how many standard deviations an element is from the mean. In the context of confidence intervals, the z-score determines how far the interval extends from the sample mean. It is linked to the desired level of confidence, which represents the probability that the interval will contain the true population parameter.

For commonly used confidence levels, the z-scores are known: for example, a 95% confidence level corresponds to a z-score of approximately 1.96. These z-scores are based on the standard normal distribution and are crucial in calculating the margin of error for the interval. Higher confidence levels would require higher z-scores, resulting in wider intervals. Researchers often balance the need for a high level of confidence with the practicality of a not-too-wide interval, making the choice of z-score a significant strategic decision in data analysis.